c space

In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences (x_n) of real numbers or complex numbers. When equipped with the uniform norm:

${\displaystyle \|x\|_{\infty }=\sup _{n}|x_{n}|}$

the space c becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, ℓ^∞, and contains as a closed subspace the Banach space c₀ of sequences converging to zero. The dual of c is isometrically isomorphic to ℓ¹, as is that of c₀. In particular, neither c nor c₀ is reflexive.

In the first case, the isomorphism of ℓ¹ with c* is given as follows. If (x₀,x₁,...) ∈ ℓ¹, then the pairing with an element (y₁,y₂,...) in c is given by

${\displaystyle x_{0}\lim _{n\to \infty }y_{n}+\sum _{i=1}^{\infty }x_{i}y_{i}.}$

This is the Riesz representation theorem on the ordinal ω.

For c₀, the pairing between (x_i) in ℓ¹ and (y_i) in c₀ is given by

${\displaystyle \sum _{i=0}^{\infty }x_{i}y_{i}.}$

References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.